JEE Main 2022 (Online) 28th July Morning ShiftQuestion 1
Let $P(-2,-1,1)$ and $Q\left(\frac{56}{17},\frac{43}{17},\frac{111}{17}\right)$ be the vertices of the rhombus $PRQS$. If the direction ratios of the diagonal $RS$ are $\alpha,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2+\beta^2$ is equal to ________.
3D Geometry
Answer$450$
Solution
In a rhombus the diagonals are perpendicular, so $RS\perp PQ$. $\overrightarrow{PQ}=\frac1{17}(90,60,94)\parallel(45,30,47)$. $45\alpha-30+47\beta=0\Rightarrow 45\alpha+47\beta=30$. General integer solution: $\alpha=-15+47s,\ \beta=15-45s$. Minimum absolute values at $s=0$: $\alpha=-15,\beta=15$. $\alpha^2+\beta^2=225+225=450$.